[PDF] [PDF] 5 Euler Characteristic - Linda Green

5 Topology

The torus and sphere are examples of closed surfaces The Möbius band and cylinder have some things in common such as that the Euler characteristic is zero 

[PDF] 5 Euler Characteristic - Linda Green

Question Is it possible to compute Euler characteristic for surfaces with boundary ? Find the Euler characteristic of: • A disk • A cylinder • A Mobius band 91 

[PDF] surfaces, which are topological spaces that

piecewise linear techniques and with the help of the Euler characteristic 5 3 1 Torus 1 0 0 0 yes Klein bottle 0 2 0 0 no Möbius strip 0 1 1 0 no Cylinder

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10 déc 2016 · The cylinder is not a surface: We usually say the cylinder is a I said the Euler characteristic is the same for homeomorphic surfaces, it better

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Now some surfaces (e g , a plane, an infinite cylinder) don't have a boundary because any boundary would be infinitely far away But other, more “finite- looking” 

[PDF] Manifolds Euler Characteristic

Only one cut makes it a cylinder which again can be quantity v−e+f is called the Euler characteristic of the surface which is a topological invariant It 1Note by  

[PDF] 1 Euler characteristics

Prove that the value of the Euler characteristic χ(S2) = V − E + F in Problem 1 The cylinder X is obtained from the unit square [0,1] × [0,1] by making the identi- 

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with boundary are the (closed) disk, the cylinder, and the Möbius strip, all Euler characteristic is independent of the triangulation for every 2-manifold

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24 nov 2011 · Example 4 The Euler characteristic χ(M) ∈ Z of a manifold M is a topological of a Möbius band is a cylinder 1 S x I = M(0,2) M = N(1,1) 

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§5EULERCHARACTERISTIC

§5EulerCharacteristic

Thegoalsfor thispartar e

•tocalculatethe Eulercharacteristic ofsurfaces •touseEuler characteristictopr ovecertainconstr uctionsare impossible •touseEuler characteristicto identifysurfaces.

References:

•TheSymmetriesof ThingsbyConway, Burgiel,andGoodman-Strauss,chapter 7 •GeometryJunkyard's 20proofsofEuler 'sTheorem •CircleinaBox bySamV andervelde,Chapter5

Supplies:

•Snaptogetherpolygons •Soccerball •Zome •Balloonsandsharpies orwhiteboard markers •Criss-crossgameboards 82
Thegas, electricity,and waterproblem§5EULERCHARACTERISTIC

Thegas,electricity ,andwater problem

Supposethere arethreecottageson aplaneandeachneedsto beconnectedtothe gas,water, andelectricitycompanies.Usinga thirddimension orsendingany ofthe allnineconnections withoutanyof thelinescr ossingeachother? 83
Faces,vertices, andedgesof polyhedra§5EULERCHARACTERISTIC

Faces,vertices, andedgesof polyhedra

Apolyhedron isa3-dimensionalshapewith flatpolygonfaces, straightedges,and sharpcorners,called vertices.Forexample, acube

andatetrahedr onare polyhedra.Foreachofthesepolyhedra, countthefaces,edges,andvertices.

ObjectFaces(F)Edges(E) Vertices(V)

Cube

Tetrahedron

Octahedron

Dodecahedron

Icosohedron

Prismon n-sidedbase

Pyramidonn-sidedbase

PentagonalCupola

SoccerBall

•Findaformula relating thenumberof faces,edges,andverticesofa polyhedra. •Thisformulais knownas Euler'sformula. 84
Euler'sfomulaonthe sphereand ontheplane §5EULER CHARACTERISTIC

Euler'sfomulaonthesphereand ontheplane

•Note:We don'treallyneedthefaces tobeflat ortheedgestobestraight tocount them.We couldimagineblowingairinto apolyhedron andpu!ngitup likea balloonandwe couldstillcount faces,edges,and verticesonthe balloonsurface. •Anetworkof vertices,edges,and facesonthe sphereis calledamaponthe sphere. •Trytobreak Euler'sformula: drawamaponthespher eforwhichitdoesnot hold. •Whatconditionsdo weneedon thefacesand theedgesto makesure thatEuler 's formulaalwaysholds forany mapona sphere? 85
Euler'sfomulaonthe sphereand ontheplane §5EULER CHARACTERISTIC thepolyhedrais drawnona flexiblerubber ball.Take anyfaceand puncha holein it,thenstr etchtheedges ofthatholeuntilt heholeis muchbiggerthan theoriginal polyhedra.Forexample, thiswouldturn acubeinto thefollowingfigur e.The puncturedfaceisnowthe infiniteoutsider egionofthe figure. Exercise.Representatetrahedronand yourotherfavorite polyhedrainasimilarway.

Wewon'tworryaboutstraight edgesanymore.

86
Euler'sfomulaonthe sphereand ontheplane §5EULER CHARACTERISTIC Definition.Aplanargraphisacollection ofpoints,called vertices,andline segments, callededges,drawn ontheplane, suchthat eachedgeconnects twovertices(which mightbothbe thesamevertex) andedges onlymeetat vertices(theydon't crosseach other).

Exercise.Drawafew planargraphs.

Question.DoesEuler's formulaholdforplanargraphs?

87
Gas,water ,andelectricity,revisited §5EULERCHARACTERISTIC

Gas,water ,andelectricity,revisited

WhatdoesEuler characteristichave todowith thegas,water ,andelectricity problem? 88
Gas,water ,andelectricity,revisited §5EULERCHARACTERISTIC Example.Findaway toposition fourpointson asheetof papersothat whenevery manysegmentswill therebe?) Example.Isitpossible todraw5 pointson theplaneand connecteachpair ofpoints withast raightlinesegment insuchawaythat thesegmentsdo notcross? Proveyour answer. 89
Eulercharacteristic forothersurfaces §5EULER CHARACTERISTIC

Eulercharacteristic forothersurfaces

Question.DoesEuler's formulastillholdforthe vertices,edges,and facesofapoly- hedraltorus? Question.Howcouldyou computethe Eulercharacteristic ofthetor usjustfr oma gluingdiagram? Question.Isitpossible tosolvethe gas,water, andelectricitypr oblemonthe torus? 90
Eulercharacteristic forsurfaceswith boundary§5EULERCHARACTERISTIC

Eulercharacteristic forsurfaceswith boundary

Question.Isitpossible tocomputeEuler characteristicfor surfaceswithboundary?

FindtheEuler characteristicof:

•Adisk •Acylinder •AMobiusband 91
Anelectric chargepr oofofEuler'sformulafor thesphere.§5EULERCHARACTERISTIC Anelectric chargeproofof Euler's formulaforthe sphere.

Question.WhydoesEuler 'sformulahold onthesphere?

92
Aninking-in proofof Euler'sformula§5EULERCHARACTERISTIC

Aninking-in proofofEuler 'sformula

V!E+Fusingther edlinesis thesameasV!E+Fusingtheblack lines. 93
Aninking-in proofof Euler'sformula§5EULERCHARACTERISTIC •Theinking-inpr oofworks foranytopologicalsurface,notjust asphere, toshow thatV!E+Fisthesame foranymap onthatsurface. •TheEulercharacteristic iscalled atopologicalinvariantofthesurface.

Additionalpr oofsofEuler'sformulafor thesphere canbefoundatDavidEppstein's GeometryJunkyard website.Look forTwenty

ProofsofEuler'sFormula V!E+F=2

94
ProblemsonEulerCharacteristic §5EULER CHARACTERISTIC

Problemson EulerCharacteristic

1.Ina certainsmallcountry,therearevillages,expressways,andfields.Expressways

totravelfr omanyvillage toanyothervillagealongthe expressways.Each fieldis completelyenclosedby expresswaysand villages.If thereare70 villagesand100 expressways,howmanyfieldsar ethere?

2.Isit possibletosolve thehousesand utilitiesproblem onator us?Demonstrate a

solutionorshow thatit isnotpossible.

3.Findthe Eulercharacteristic ofthefollowing surfaces:

(a)thepr ojectiveplane(Hint: workfromthegluingdiagram.) (b)theKlein bottle (c)apair ofpants

4.Apolyhedr onismade upofpentagonsandhexagons insucha waythatthr ee

polygonsmeetat eachvertex.How manypentagons mustthere be?Prove thatno othernumberof pentagonsispossible. Challenge:whatar ethepossible answers forthenumber ofhexagons?

5.Acertain polyhedroni sbuilt entirelyfromtriangles,in suchawaythat5faces

meetateach vertex.Pr ovethatit hastohave20faces.(Hint: firstdeducethat3F 95
ProblemsonEulerCharacteristic §5EULER CHARACTERISTIC =2Eand3F =5V)

6.Anotherpolyhedr onisbuiltentirelyfromtriangles,insuchawaythat4facesmeet

ateachvertex. Prove thatithas tohave8faces.Whatabout if3faces meetateach vertex?Whathappens if6faces meetateach vertex?

7.Apolyhedr oniscalled regularifallitsfaces arethe samer egularpolygons(for

example,allequilateral trianglesor allsquares) andifthe samenumberof faces meetat eachvertex(for example,3faces meetateach vertex).Forexample, the cube,thetetrahedr on,the octahedron,thedodecahedron,andtheicosahedron are allregular polyhedra,buttheshapemadeby gluingtwotetrahedra togetheralong atriangleis not,becausesome verticeshave4 trianglesaround themandothers have3.Pr ovethatthe fiveregularpolyhedramentionedin theprevious sentence aretheonlyregular polyhedrapossible.(Hint: startyourar gumentbyusingthe problemsabove.)Regularpolyhedraar ealsocalled Platonicsolids.

8.UsingEuler 'sFormula, provethatyoucan'tconnect eightpointspairwise ona

torus,insuchaway thatnoneof theconnectingcurves intersect.("Thecomplete graphK8cannot beembeddedin atorus.")

9.Connectfive pointspairwiseon atorus, insucha waythat noneofthe connecting

curvesintersect.Now dosix points;thenseven (hard!).

10.What'sthe maximumnumberof pointsthatcan beconnectedpairwise onthe

projectiveplane?OnaKlein bottle? 96
ProblemsonEulerCharacteristic §5EULER CHARACTERISTIC

11.Pr ovethatanytriangulationofthe torusmust haveatleast 7vertices.That is,your

solutiontothe previouspr oblem(connecting7 verticespairwiseonthetorus) constitutesaminimal triangulationofthe torus.What's theanalogous resultfor theprojective plane?

12.Acertain polyhedronis madeofsquar es,hexagons,anddecagons(whichar e10-

sidedpolygons),in suchaway thatonesquar e,onehexagon, andonedecagon meetateach vertex.Howmany verticesdoesit have?

13.Whatar e4-dimensionalanalogs ofcubesandotherpolyhedra?How couldyou

defineEulercharacteristic forthem?What numberdothe partssumto?

14.What patternsholdfor thenumberof faces,edges,and verticesforthese planar

graphs(alsoknown asscribbles)?(Aplanar graphisa collectionofvertices and edgesdrawnin theplane,in suchaway thatthere isavert exat thebeginningand endofeach edgeand atanyplace wheretwo edgesmeetor cross.) Notethatthe planargraph(or scribble)canhave morethan onecomponent.

15.Prove theformulayoufound forscribbles,which shouldbeageneralizat ionof

97
ProblemsonEulerCharacteristic §5EULER CHARACTERISTIC Euler'sformula.Hint:first, showthatthe formulaholdsfor ascribblewith 0 edges.Thenuse aninductionar gumentbasedon thenumberof edges.What happenstothe patternyouhave observedifyou removean edgefrom ascribble? Manyofthese problemsar efrom CircleinaBoxbySamV anderveldeandfr omBeginningTopologybySue Goodman. 98

TheGame ofCriss-Cross §5EULER CHARACTERISTIC

TheGame ofCriss-Cross

Thegameof Criss-Crossis playedona gameboardcreatedbydrawing three pointsat theverticesof alarge equilateraltriangle, alongwitht wotosevenadditionalpoints anywhereinitsinterior .Playersalternate turnsdrawinga singlestraightlinesegment joiningany twopoints,as longasthe segmentdoesnot passthrough anyotherpoints orsegmentsalr eadyappearingon thegameboard.Thewinner isthe lastplayerable tomakea legalmove. Exercise.Playsomegames ofcriss-cr osswitha classmate.Isther eawinningstrategy?

Doyouwant togofirst orsecond?Analyze thegame!

99
Eulercharacteristic ofsomefamiliar surfaces§5EULERCHARACTERISTIC

Eulercharacteristic ofsomefamiliar surfaces

FindtheEuler characteristicfor:

1.ator us

2.a2-holed torus

3.acylinder (withoutthetop orbottom)

4.acone (withoutthebottom)

5.a hexagon

6.aMobius band

Notethatthe cylinder,cone, hexagon,andMobius bandaresurfaceswithboundary. Areallintegerspossibleas theEulernumber forsurfaces? 100
Buildingnew surfacesoutof old§5EULER CHARACTERISTIC

Buildingnew surfacesoutof old

Twowaysofbuildingnew surfaces(withand withoutboundary)out ofoldar e:

1.makingpunctur es

2.joiningsurfaces withconnected sums

Question.Whatdopunctur esdoto Eulercharacteristic? Question.Whatdo connectedsumsdo toEulercharacteristic? 101
WhatEuler characteristicsare possible?§5EULERCHARACTERISTIC

WhatEuler characteristicsarepossible?

Forsurfaceswithout boundary?

S 2 T 2 P 2 T 2 #T 2 P 2 #P 2 T 2 #T 2 #T 2 P 2 #P 2 #P 2 T 2 #T 2 #T 2 #T 2 P 2 #P 2 #P 2 #P 2

Forsurfaces withboundary?

102
Classifysurfaces byEulercharacteristic (andorientabililty)§5EULERCHARACTERISTIC Classifysurfaces byEulercharacteristic (andorientabililty)

Whatsurfaceshave

•Eulercharacteristicof 2? •Eulercharacteristicof 1? •Eulercharacteristicof 0? •Eulercharacteristicof -1? 103

Whatsurface isthis?§5EULER CHARACTERISTIC

Whatsurface isthis?

Exercise.Whattopologicalsurfaces doeachof thesefigures represent? UseEuler characteristic,orientabiity ,andnumberofboundarycir clestodecide. 104

Whatsurface isthis?§5EULER CHARACTERISTIC

Singleelimination gluingdiagramidentification tournament.Seewho cancorrectly identifythesurface fromthe gluingdiagramfirst. Thelosergetstodrawthe gluing diagramforthe nextpairof contestants. 105
Problemsonidentifyingsurfaces usingEulercharacteristic §5EULER CHARACTERISTIC Problemson identifyingsurfacesusing Eulercharacteristic

1.Identifythese surfaces.

2.Identifythis surfacethat isahole withinahole withinahole.

Holeina holeina hole

Seealso theNumberphilevideo.

3.Take apieceofpaperinthe shapeofa cross,and identifyoppositeends Ther esult

isasurface withboundary. Howmanyboundary componentsare there?Ifyou weretodescribether esultasa closedsurfaceMwithnpunctures,whatwouldM 106
Problemsonidentifyingsurfaces usingEulercharacteristic §5EULER CHARACTERISTIC beand howmanypunctur es?Ifyour answerdependson thegluinginstructions fortheends besure tocoverall cases.

4.Identifythe surfacesobtainedby

(a)Gluing theoppositesides ofanoctagon asanorientable surface. (b)Gluingthe oppositesides ofadecagon asanorientable surface. (c)Gluingthe sidesofa polygonwithgluing instructionsgiven bythewor d b 1 a 2 b 2 a !1 3 b !1 2 a 3 a !1 2 a !1 1 b !1 1 a 1 (d)Gluingthe sidesof apolygonwith gluinginstructions givenbythe word b !1 1 a 2 b 2 a !1 3 b !1 2 a 3 a !1 2 a !1 1 b !1 1 a 1 (e)Gluingtwo polygonswithwordsaroundtheoutsideof:ad !1 e !1 fga !1 b !1 c !1 and debf !1 cg

5.Ahexagon hasits edgesidentifiedin pairsofletters a,b,ctogivea surface.How

manydistinct(not topologicallyequivalent) surfacescanbe represented inthis way?Givea wordr epresentingeach suchsurface.Hint:Beginbyfinding upper andlowerbounds fortheEuler characteristic,thusr educingtoa smallnumberof casesforwhich youneedto constructan example.

6.We haveseenthatM#S

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